Problem 12 - Brief Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. \(\left(-2, \frac{1}{2}\right)\) and \(\left(5, \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The slope is 0 (the line is horizontal).

Step 1: Identifying Coordinates

The given points are \((-2, \frac{1}{2})\) and \((5, \frac{1}{2})\). Let's label these points as \((x_1, y_1) = (-2, \frac{1}{2})\) and \((x_2, y_2) = (5, \frac{1}{2})\).

Step 2: Applying the Slope Formula

The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as follows:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 3: Substituting the Values

Substitute the appropriate values into the slope formula:\[m = \frac{\frac{1}{2} - \frac{1}{2}}{5 - (-2)} = \frac{0}{5 + 2}\]

Step 4: Simplifying the Expression

Simplify the expression:\[m = \frac{0}{7} = 0\]Since the numerator is 0, the slope is 0, which means the line is horizontal.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This type of geometry allows us to use algebra to solve geometric problems by representing points using coordinates. In a two-dimensional plane, each point has an \( x \) (horizontal) coordinate and a \( y \) (vertical) coordinate, which we write as \((x, y)\).

• The x-coordinate tells you how far left or right the point is located.
• The y-coordinate tells you how far up or down the point is located.

Coordinate geometry makes it easy to calculate distances, slopes, and other important properties of geometric figures. When you have two points, as in the exercise \((-2, \frac{1}{2}\) and \(5, \frac{1}{2}\)), you can determine the nature of the line passing through them by applying concepts from this field.

Slope Formula

The slope of a line is a measure of its steepness and direction. In coordinate geometry, we calculate the slope using the slope formula. It provides a numerical value that describes how one variable changes in relation to another.

The slope \( m \) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

- If the slope is positive, the line rises as it moves from left to right.- If the slope is negative, the line falls as it moves from left to right.- A zero slope indicates a horizontal line, which does not rise or fall as you move along it.
In our problem, substituting the values of \(y_1\) and \(y_2\) gives \( \frac{(\frac{1}{2} - \frac{1}{2})}{(5 - (-2))} = \frac{0}{7} = 0 \). This zero slope is typical of horizontal lines.

Horizontal Line

A horizontal line is a straight line that runs left to right across the coordinate plane. In terms of coordinate geometry, a horizontal line has a slope of zero. This is because the \( y \)-coordinates of any two points on a horizontal line are the same, leading the slope formula's numerator to be zero.

- Key characteristics of a horizontal line: - Constant \( y \)-value for any \( x \)-value. - Appears flat on the graph, with no incline or decline.

In the provided example with points \((-2, \frac{1}{2})\) and \(5, \frac{1}{2}\), the \( y \)-coordinates are equal. This confirms that the line is horizontal and exemplifies the nature of such lines in coordinate geometry. Horizontal lines often appear in real-world scenarios where a specific value remains consistent across various situations, such as water levels in a tank at capacity.